Problem Description:
The task involves analyzing a scenario where a company is considering whether to implement a new software system. The decision depends on the outcomes of two independent events:
Event A (whether the current system will fail within the next year) and
Event B (whether the new system will be successfully implemented). The probabilities of these events are given, and the goal is to determine the probability that the company will need to make a purchase decision based on the outcomes of these events.
Key Information from the Image:
1.
Event A: The current system fails within the next year.
- Probability of Event A occurring: \( P(A) = 0.3 \)
- Probability of Event A not occurring: \( P(A^c) = 1 - P(A) = 0.7 \)
2.
Event B: The new system is successfully implemented.
- Probability of Event B occurring: \( P(B) = 0.6 \)
- Probability of Event B not occurring: \( P(B^c) = 1 - P(B) = 0.4 \)
3.
Decision Rule:
- The company needs to make a purchase decision if:
- The current system fails (
Event A occurs), or
- The new system is not successfully implemented (
Event B does not occur).
Solution Approach:
To solve this problem, we need to calculate the probability that the company will need to make a purchase decision. This happens under the following conditions:
-
Event A occurs, or
-
Event B does not occur.
Mathematically, this can be expressed as:
\[
P(\text{Purchase Decision}) = P(A \cup B^c)
\]
Using the principle of inclusion-exclusion for probabilities:
\[
P(A \cup B^c) = P(A) + P(B^c) - P(A \cap B^c)
\]
#### Step 1: Calculate \( P(A) \) and \( P(B^c) \)
From the given information:
\[
P(A) = 0.3
\]
\[
P(B^c) = 0.4
\]
#### Step 2: Calculate \( P(A \cap B^c) \)
Since Events A and B are independent, the probability of both events occurring together is the product of their individual probabilities:
\[
P(A \cap B^c) = P(A) \cdot P(B^c) = 0.3 \cdot 0.4 = 0.12
\]
#### Step 3: Apply the Inclusion-Exclusion Formula
Substitute the values into the formula:
\[
P(A \cup B^c) = P(A) + P(B^c) - P(A \cap B^c)
\]
\[
P(A \cup B^c) = 0.3 + 0.4 - 0.12 = 0.58
\]
Final Answer:
The probability that the company will need to make a purchase decision is:
\[
\boxed{0.58}
\]
Explanation:
-
Event A (current system failure) alone contributes to the need for a purchase decision.
-
Event B^c (new system not being successfully implemented) also contributes independently.
- Since the events are independent, we use the inclusion-exclusion principle to avoid double-counting the scenario where both events occur simultaneously.
- The final probability reflects the combined likelihood of either event occurring, ensuring the company must make a purchase decision.
Parent Tip: Review the logic above to help your child master the concept of hypothesis worksheet.
